When does \(SC(X)=\mathbb R^X\) hold?
type: Item
instance of: scholarly article
title: When does \(SC(X)=\mathbb R^X\) hold?
zbMATH Open document ID: 1210.54046
author: B. M. Bokalo
author: Nadiya Kolos
published in: Topology
publication date: 2010-05-10T00:00:00Z
review text: A map \(f:X\rightarrow Y\) between topological spaces is called scatteredly continuous if for each non-empty subspace \(A\subset X\) the restriction \(f|_A\) has a point of continuity. Let \(SC(X)\) denote the set of all scatteredly continuous maps from \(X\) to the space of real numbers \(\mathbb R\). The authors consider the problem: what conditions on \(X\) imply the equality \(SC(X)=\mathbb R^X\)? The main results: (1)~If \(SC(X)=\mathbb R^X\) then every countable subspace of \(X\) is scattered. (2)~If \(X\) has countable tightness, then \(SC(X)=\mathbb R^X\) iff \(X\) is scattered. (3)~Assume \(V=L\). Then \(SC(X)=\mathbb R^X\) iff \(X\) is scattered. (4)~Assume that \(ZFC\) and \textit{there exists a measurable cardinal} is consistent. Then there exists a completely regular \(0\)-dimensional crowded space \(X\) such that \(SC(X)=\mathbb R^X\).
reviewed by: Tomasz Natkaniec
Mathematics Subject Classification ID: 54G12
Mathematics Subject Classification ID: 54C08
Mathematics Subject Classification ID: 26A21
Mathematics Subject Classification ID: 54C10
zbMATH DE Number: 5706259
zbMATH Keywords: scatteredly continuous map
zbMATH Keywords: resolvable space
zbMATH Keywords: irresolvable space
zbMATH Keywords: dividing and undividing sets
zbMATH Keywords: scattered space
zbMATH Keywords: \(Q\)-space
MaRDI profile type: Publication
full work available at URL: j.top.2009.11.016
OpenAlex ID: W194974792
DOI: 10.1016/J.TOP.2009.11.016
description: scientific article; zbMATH DE number 5706259